On 14/08/2013 1:20 AM, quasi wrote:> Nam Nguyen wrote:>> quasi wrote:>>> Nam Nguyen wrote:>>>> quasi wrote:>>>>> Nam Nguyen wrote:>>>>>> quasi wrote:>>>>>>> Nam Nguyen wrote:>>>>>>>> quasi wrote:>>>>>>>>> Nam Nguyen wrote:>>>>>>>>>>>>>>>>>>>> More than once, I was asked what the difference between the>>>>>>>>>> Goldbach Conjecture and its weaker form that an odd number>>>>>>>>>> greater than 7 is the sum of three odd primes.>>>>>>>>>>>>>>>>>>>> The point is though the essences of the 2 conjectures are>>>>>>>>>> drastically different: _an odd number can not be defined>>>>>>>>>> without addition_ while an even number can (as per Def-03b>>>>>>>>>> above).>>>>>>>>>>>>>>>>>> Consider the following statements:>>>>>>>>>>>>>>>>>> GC_2: All sufficiently large even numbers can be expressed as>>>>>>>>> the sum of 2 primes.>>>>>>>>>>>>>>>>>> GC_4: All sufficiently large even numbers can be expressed as>>>>>>>>> the sum of 4 primes.>>>>>>>>>>>>>>>>>> GC_6: All sufficiently large even numbers can be expressed as>>>>>>>>> the sum of 6 primes.>>>>>>>>>>>>>>>>>> etc ...>>>>>>>>>>>>>>>>>> It seems you claim to have proved:>>>>>>>>>>>>>>>>>> "It impossible to know whether or not GC_2 is true.">>>>>>>>>>>>>>>> That's not what I claimed ...>>>>>>>>>>>>>>>> ... my much more restricted claim to prove (as opposed to my>>>>>>>> own about cGC) would be:>>>>>>>>>>>>>>>> (*) If the Goldbach conjecture is true in the natural numbers,>>>>>>>> then it's impossible to structure theoretically prove, verify>>>>>>>> it so.>>>>>>>>>>>>>> OK.>>>>>>>>>>>>>>>> Would your proof method also suffice to prove the same for>>>>>>>>> GC_4? For GC_6? etc?>>>>>>>>>>>>>>>> Before this, do you agree there has been a misunderstanding>>>>>>>> on what I had said to Virgil I could prove here, in>>>>>>>> relation to GC_2?>>>>>>>>>>>>>> Sure, no problem -- agreed.>>>>>>>>>>>>>>> If you do agree, would your questions about GC_4 and GC_6>>>>>>>> _still_ stand?>>>>>>>>>>>>>> Yes.>>>>>>>>>>>> Then the answer is No: my proof wouldn't be sufficient for>>>>>> GC_4 or GC_6 (should they be true), but for a different>>>>>> reason compared to the weak Goldbach Conjecture.>>>>>>>>>> What reason?>>>>>>>> Please see below.>>>>>>>>>> Note that in the case of G_4,>>>>>> (sum of 4 primes) =>(sum of 2 evens);>>>>>> and in the case of G_6,>>>>>> (sum of 6 primes) => (sum of 2 odds).>>>>>>>>>> Is the above sentence supposed to be the reason why your proof>>>>> method can't generalize from G_2 to G_4 or G_6?>>>>>>>> Yes.>>>>>>>> As I've alluded to in my recent response to you, the conclusion>>>> of GC_2 isn't of the same essence as those of GC_4 and GC_6.>>>>>> The conclusions are clearly not the same.>>>>>> After all, the numbers 2,4,6 are not the same.>>>>>> The question I asked was whether the proof technique you used>>> to show that GC_2, if true, is not provable, would generalize>>> to show the same for GC_4 or GC_6. The conclusions, while>>> different, have some similarities, so it's not inconceivable>>> that you could apply essentially the same reasoning for GC_4>>> or GC_6 as you did for GC_2.

The problem is "have some similarities" is provably so vague that itwon't offer help in comparison, while "isn't of the same essence"is actually defensible.

For instance, let's consider:

GC_7: All sufficiently large even numbers can be expressed as the sum of 6 primes, which is zero.

Obviously GC_7 is false but it definitely "has some similarities" sinceit's still about the sum of (6) primes.

On the other hand, a _sum of exactly 2 primes_ (Goldbach Conjecture,GC_2) means we're talking about addition function reduction overthe sub-domain _consisting of primes only_ , while in the case of"sum of 4 primes" or "sum of 6 primes" the sub-domain for additionwould contain no primes, or would contain more than just primes.

So: different essence.

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